***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St000139 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The Coxeter number of a finite Cartan type. The Coxeter number $h$ for the Weyl group $W$ of the given finite Cartan type is defined as the order of the product of the Coxeter generators of $W$. Equivalently, this is equal to the maximal degree of a fundamental invariant of $W$, see also [[St000138]]. ----------------------------------------------------------------------------- References: [1] Humphreys, J. E. Reflection groups and Coxeter groups [[MathSciNet:1066460]] ----------------------------------------------------------------------------- Code: def statistic(cartan_type): return prod(WeylGroup(cartan_type).gens()).order() ----------------------------------------------------------------------------- Statistic values: ['A',1] => 2 ['A',2] => 3 ['B',2] => 4 ['G',2] => 6 ['A',3] => 4 ['B',3] => 6 ['C',3] => 6 ['A',4] => 5 ['B',4] => 8 ['C',4] => 8 ['D',4] => 6 ['F',4] => 12 ['A',5] => 6 ['B',5] => 10 ['C',5] => 10 ['D',5] => 8 ['A',6] => 7 ['B',6] => 12 ['C',6] => 12 ['D',6] => 10 ['E',6] => 12 ['A',7] => 8 ['B',7] => 14 ['C',7] => 14 ['D',7] => 12 ['E',7] => 18 ['A',8] => 9 ['B',8] => 16 ['C',8] => 16 ['D',8] => 14 ['E',8] => 30 ['A',9] => 10 ['B',9] => 18 ['C',9] => 18 ['D',9] => 16 ['A',10] => 11 ['B',10] => 20 ['C',10] => 20 ['D',10] => 18 ----------------------------------------------------------------------------- Created: Jun 24, 2013 at 12:53 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Jun 01, 2015 at 17:58 by Martin Rubey